Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500839 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#2250 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
657 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,11)(8,9)(10,12)(14,20)(15,26)(16,22)(17,28)(18,21)(23,24)(25,27);(1,2)(3,4)(6,9)(10,11)(13,14,17,23,18,24,16,20,19,25,22,15,21,26,28,27);(2,4)(13,15,16,14,21,20,17,26,19,23,28,25,18,27,22,24);(5,7,8,12)(6,10,9,11)(13,14,22,15,18,24,28,27,19,25,17,23,21,26,16,20)Polycyclic presentation:
2^9|b^2=f,c^2=fh,d^2=eh,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,h^a=hi,c^b=cg,d^b=di,d^c=di74 conjugacy classes — view the full character table.
| maximal subgroup of | #72319014, #72319486, #72319607 |
| maximal normal subgroup of | #72319014, #72319486, #72319607 |
| maximal quotient of | #72322603 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 411198);